A segment has a midpoint #(-2, 9)# and one endpoint #(2,8)#. What is the coordinate of the other endpoint?

2 Answers
Jun 7, 2017

Answer:

See a solution process below:

Explanation:

The formula to find the mid-point of a line segment give the two end points is:

#M = ((color(red)(x_1) + color(blue)(x_2))/2 , (color(red)(y_1) + color(blue)(y_2))/2)#

Where #M# is the midpoint and the given points are:

#(color(red)(x_1, y_1))# and #(color(blue)(x_2, y_2))#

Subtituting gives:

#(-2, 9) = ((color(red)(2) + color(blue)(x_2))/2 , (color(red)(8) + color(blue)(y_2))/2)#

Solving for #x# gives:

#-2 = (color(red)(2) + color(blue)(x_2))/2#

#color(green)(2) xx -2 = color(green)(2) xx (color(red)(2) + color(blue)(x_2))/2#

#-4 = cancel(color(green)(2)) xx (color(red)(2) + color(blue)(x_2))/color(green)(cancel(color(black)(2)))#

#-4 = color(red)(2) + color(blue)(x_2)#

#-2 - 4 = -2 + color(red)(2) + color(blue)(x_2)#

#-6 = 0 + color(blue)(x_2)#

#-6 = color(blue)(x_2)#

#color(blue)(x_2) = -6#

Solving for #y# gives:

#9 = (color(red)(8) + color(blue)(y_2))/2#

#color(green)(2) xx 9 = color(green)(2) xx (color(red)(8) + color(blue)(y_2))/2#

#18 = cancel(color(green)(2)) xx (color(red)(8) + color(blue)(y_2))/color(green)(cancel(color(black)(2)))#

#18 = color(red)(8) + color(blue)(y_2)#

#-8 + 18 = -8 + color(red)(8) + color(blue)(y_2)#

#10 = 0 + color(blue)(y_2)#

#10 = color(blue)(y_2)#

#color(blue)(y_2) = 10#

The other end point is: #(color(blue)(-6, 10))#

Jun 7, 2017

Answer:

The coordinate of the other endpoint is #(-6, 10)#

Explanation:

Instead of full-on visualizing a graph, let's start with a number line:

This is a simple number line from #0# to #10#. Now, I ask you, what is an easy way to calculate the midpoint between #0# and #10#? Of course it's #5# but is there a formula we can derive to calculate the midpoint between any two points? Well, let's see. If we took #0# and #10#, added them together, then divided the quotient by #2#, boom! We get #5#!

Let's try it with two more numbers. How about #2# and #4#? If we were to add the two up, then divide the result by #2#, we'd get #3# - also the midpoint between #2# and #4#.

Now we can see that there's a formula that we can use to calculate the midpoint between any two numbers! Add the two end numbers up and divide the result by #2#! This formula - known as the Midpoint Formula - is shown below:

For any two endpoints #x_1# and #x_2# on the number line,

#M=(x_1+x_2)/2#

So how does all of this relate to the problem? Well, remember that the graph is a coordinate plane, made up of two axes - the x-axis and the y-axis. And you can think of each axis to be a number line!

So now, our task is to derive a formula for finding the midpoint between any two points on the coordinate point. That way, we can write a relationship between the two endpoints on the coordinate plane and the midpoint between those two points.

Suppose there was a graph like the one shown below:

Suppose point A had the coordinates #(x_1, y_1)# and point B had the coordinates #(x_2, y_2)#. And suppose there was also a point M that was the midpoint between points A and B. Now, we want to write the midpoint's coordinates in terms of the #x# and #y# coordinates of points A and B. That way, we can connect the coordinates of points A and B with the coordinates of their midpoint. Now, the coordinates of the midpoint are essentially the midpoint between the two x-coordinates and the midpoint between the two y-coordinates.

Let's focus on the x-axis first and treat it as a number line, except without numbers. On the x-axis, we have the x-coordinate of point A as well as the x-coordinate of point B. From our earlier rule, we can state that the midpoint between the two x-coordinates is:

#M=(x_1+x_2)/2#

Likewise, the midpoint between the two y-coordinates is:

#M=(y_1+y_2)/2#

As we stated earlier, the coordinates of the midpoint between points A and B are the midpoint between the two x-coordinates and the midpoint between the two y-coordinates. Combining the two statements above, we get the conclusion that the coordinates of the midpoint between points A and B are:

#M=((x_1+x_2)/2, (y_1+y_2)/2)#

Now, with that information, we can substitute in the values mentioned in the question. We let point A and its coordinates be our missing endpoint, point B and its coordinates be our known endpoint, and point M, our midpoint, and its coordinates be the midpoint, like so:

#M(-2, 9)=((x_1+2)/2, (y_1+8)/2)#

This equation tells us that #-2=(x_1+2)/2# and that #9=(y_1+8)/2#. Next, we start solving the equations to figure out what #x_1# and #y_1# are.

Let's start with the left equation:

#-2=(x_1+2)/2#

#x_1+2=-4#

#x_1=-6#

And now, let's solve the right equation:

#9=(y_1+8)/2#

#y_1+8=18#

#y_1=10#

And now, we put the two values together to form the answer:

#(-6, 10)#